What is the moment of inertia of a square sheet of side `l` and mass per unit area `mu` about an axis passing through the centre and perpendicular to its plane

Options

A. `(mul^(2))/(12)`
B. `(mul^(2))/(6)`
C. `(mul^(4))/(12)`
D. `(mul^(4))/(6)`

Answer

To find the moment of inertia of a square sheet of side \( l \) and mass per unit area \( \mu \) about an axis passing through the center and perpendicular to its plane, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Area of the Square Sheet**:
The area \( A \) of the square sheet can be calculated using the formula:
\[ A = l^2 \]

2. **Calculate the Total Mass of the Square Sheet**:
Given the mass per unit area \( \mu \), the total mass \( M \) of the square sheet can be calculated as:
\[ M = \mu \times A = \mu \times l^2 \]

3. **Use the Formula for Moment of Inertia**:
The moment of inertia \( I \) of a square sheet about an axis passing through its center and perpendicular to its plane is given by the formula:
\[ I = \frac{1}{6} M l^2 \]

4. **Substitute the Total Mass into the Moment of Inertia Formula**:
Now, substituting the expression for \( M \) into the moment of inertia formula:
\[ I = \frac{1}{6} (\mu l^2) l^2 \]

5. **Simplify the Expression**:
Simplifying the equation gives:
\[ I = \frac{1}{6} \mu l^4 \]

### Final Result:
Thus, the moment of inertia of the square sheet about the specified axis is:
\[ I = \frac{\mu l^4}{6} \]